Write $(-1,1,2,2)^T$ in terms of the basis $U$

Question

I have found that $U=\{1/2\pmatrix{1\\1\\1\\1},1/2\pmatrix{-1\\-1\\1\\1},\frac{1}{\sqrt{2}}\pmatrix{-1\\1\\0\\0},\frac{1}{\sqrt{2}}\pmatrix{0\\0\\1\\-1}\}$ is an orthogonal basis for $\mathbb{R}^4$.

Now I want to write $(-1,1,2,2)^T$ as a linear combination of the elements of $U$.

As far as I can see, I can now write that

$\frac{\alpha_1}{2}\pmatrix{1\\1\\1\\1}+\frac{\alpha_2}{2}\pmatrix{-1\\-1\\1\\1}+\frac{\alpha_3}{\sqrt{2}}\pmatrix{-1\\1\\0\\0}+\frac{\alpha_4}{\sqrt{2}}\pmatrix{0\\0\\1\\-1}=\pmatrix{-1\\1\\2\\2}$.

But I still need to determine the alphas, so I see that I have four equations with four unknowns, which I solve with ERO's:

$\pmatrix{1 & -1 & -1 & 0 & -1\\1 & -1 & 1 & 0 & 1\\1 & 1 & 0 & 1 & 2\\1 & 1 & 0 & -1 & 2}\sim\pmatrix{1 & 0 & 0 & 0 & 0\\0 & 1 & 0 & 0 & 0\\0 & 0 & 1 & 0 & 0\\0 & 0 & 0 & 1 & 0}$.

Now I get that $\alpha_1=1,\alpha_2=1,\alpha_3=1,\alpha_4=0$. So

$(\frac{1}{2}\pmatrix{1\\1\\1\\1}+\frac{1}{2}\pmatrix{-1\\-1\\1\\1}+\frac{1}{\sqrt{2}}\pmatrix{-1\\1\\0\\0}+\frac{0}{\sqrt{2}}\pmatrix{0\\0\\1\\-1}=\pmatrix{-1\\1\\2\\2}$.

But this seems to be wrong. Where did I go wrong?

EDIT: I think I wrote out the alphas wrong. I think they are $\alpha_1=2,\alpha_2=2,\alpha_3=\sqrt{2},\alpha_4=0$. Is this correct?

Answer 1

It is much easiar to use the following method: if $\{e_1,,e_2,\ldots,e_n\}$ is an orthonormal basis of a vector space $V$ endowed with an inner product $\langle\cdot,\cdot\rangle$ and if $v\in V$, then if you want to write $v$ as $\alpha_1e_1+\alpha_2e_2+\cdots+\alpha_ne_n$, all you have to do is to take $\alpha_k=\langle v,e_k\rangle$, for each $k\in\{1,2,\ldots,n\}$.

Answer 2

If you want to do it that way, you need to include the scalars out in front in your matrix.

You obviously made a mistake, because your row-reduced matrix reads $\alpha_1=\alpha_2=\alpha_3=\alpha_4=0$.

Back to my first point, the matrix to row-reduce is: $\begin{pmatrix}\dfrac 12&-\dfrac 12&-\dfrac1{\sqrt2}&0&-1\\\dfrac 12&-\dfrac 12&\dfrac 1{\sqrt2}&0&1\\\dfrac12&\dfrac 12&0&\dfrac 1{\sqrt2}&2\\\dfrac12 &\dfrac12 &0&-\dfrac 1{\sqrt2}&2\end{pmatrix}$.